The number of the minimal models for a 3-fold of general type is finite (Q1078276)

In this paper we show that for a normal quasi-projective variety X of dimension 3 with only canonical singularities, the number of the following set \(M:=\{f:Y\to X;\) f is a projective, birational, and crepant morphism from a normal variety Y with only canonical singularities\} is finite. This result gives an affirmative answer to the conjecture of M. Reid and V. V. Shokurov, i.e., any 3-fold of general type has at most a finite number of minimal models. With the notion of the log-flips developed by the first author, we reveal the fact that the finiteness of the minimal models only reflects the discreteness of the extremal rays.